Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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2
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0answers
31 views

Fixed point set of orthogonal Transformation

I need some help with this problem. Let $g$ be an element of the orthogonal group and $s$ a reflection. Then the dimension of the fixed point set of $g$ and $gs$ differ by $\pm 1$. Since that ...
4
votes
1answer
91 views

groups of order $p^5$ and exponent p

We know that a group of order $p^5$ is an extra special group. How we can show it doesn't have any abelian subgroup of order $p^4$? Also what is the presentation of this group if its exponent is equal ...
1
vote
1answer
56 views

Generating a subgroup

While getting ready for my exams I am trying to solve this question "Find a Sylow $2$-subgroup in $S_4$ which contains the element $(1, 3, 2, 4)$." I started by permutation of the element required ...
-1
votes
0answers
30 views

Group Structure on the set of matrices over finite field

Consider a set $M_n$ of all possible square matrices of dimension $n$ over a finite field $F_q$. Clearly the cardinality of the set $M_n$ is $q^{n^2}$. You must be agreed with me that the set $M_n$ ...
3
votes
3answers
50 views

Is a finite group with a certain automorphism must be abelian

Let $G$ be a finite group, and let $f:G \rightarrow G$ be an automorphism, such that $x f(x) f(f(x))=1$ for any $x \in G$. Is $G$ must be abelian? I believe that there are examples where $G$ is not ...
2
votes
2answers
24 views

A group of order $56$ with a unique Sylow $2$-group is either nilpotent or its Sylow $2$-group is $\cong (\mathbb{Z}/2 \mathbb{Z})^3$

Let $G$ be a group of order $56$, $Q$ a normal Sylow $2$-group and $P$ a Sylow $7$-group. Show either $G \cong Q \times P$ or $Q \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \times ...
1
vote
0answers
31 views

For $k$ random perms of an $n$-set $\mathrm{Pr}[\sigma_1\cdots\sigma_k=\sigma_k\cdots\sigma_1]\xrightarrow{k\rightarrow\infty}\frac{2}{n!}$?

Q. Fix $n \geq 2$, and choose $k$ random permutations $\sigma_1\sigma_2\cdots\sigma_k \in S_n$ uniformly at random. Is true that ...
11
votes
2answers
143 views

Probability of $\alpha\beta\gamma=\gamma\beta\alpha$ for random permutations of a finite set?

Following up on my previous question Probability that two random permutations of an $n$-set commute?, here's a related question for three elements. Q: If $\alpha,\beta,\gamma$ are chosen uniformly ...
0
votes
1answer
47 views

Groups of order 8 proof

I understand the solution to these questions I was just wanting to confirm that the solution to Q10 excludes the possibility that $y =x^2$ ( and hence the proof is not complete) as it uses the ...
1
vote
1answer
46 views

Commutators Calculus

I was trying to understand the above Corollary but I have a problem, namely why in the second to last line $A_0 \leq \zeta_p(G)$? Any ideas? Definitions By recurrence we define $[x,_0\, y]=x$; ...
5
votes
1answer
65 views

Prove that $G$ has a nontrivial normal subgroup

I wanted to ask if I had done this problem correctly. Let $G$ be a group of order $pqr$ (for $p > q > r$ primes). (i) If $G$ fails to have a normal subgroup of order $p$, determine the ...
0
votes
0answers
16 views

Constructing all inequivalent faithful irreducible projective representations of finite abelian groups

Let $G$ be a finite abelian group, and $\alpha \in H^2 (G,\mathbb{C}^*)$ a 2-cohomology class. It is known (in Karpilovsky's multi-volume tome or elsewhere) that a finite abelian group admits a ...
1
vote
0answers
47 views

Normally embedded subgroups reducing in a Hall system

A Hall system of $G$ is a set $\Sigma$ of Hall subgroups of $G$ satisfying the following two properties: -For each $\pi$ divisor if $|G|$, $\Sigma$ contains excatly one Hall $\pi$-subgroup $G_{\pi}$. ...
4
votes
2answers
139 views

Is a finite group always a element-wise product of Sylow subgroups?

Let $G $ be a finite group, and let $ p_1, \ldots, p_n $ be the distinct primes dividing $|G|$. For each $i $, let $ P_i $ be a Sylow $ p_i $-subgroup of $ G $. I seem to recall a theorem saying $ ...
2
votes
1answer
20 views

Hopefully easy Lang-Steinberg computation: for Weyl elements

How do you write an element of the Weyl group as $g^{-1} F(g)$? For instance, let $G = \langle x_1(t), x_2(t), x_{-1}(t), x_{-2}(t) : t \in K \rangle$ where $K$ is an algebraically closed field ...
3
votes
1answer
65 views

Permutations: If I know $\alpha$ and the cycle structure of $\alpha\beta$, can I find $\gamma$ for which $\gamma\beta$ also has this cycle structure?

Suppose we have two permutations $\alpha$ and $\beta$ (of a set $S$ of size $|S|=n$), and I know $\alpha$ and the cycle structure of $\alpha\beta$. But I don't know $\beta$. Can I find a ...
1
vote
2answers
39 views

Operations in finite field $F_p$

If I have a finite field $F_p$, where $p$ is prime how can I define operations like $+, -, \times, / $? Can I just make: $$add: (a + b) \mod p$$ $$sub: (a-b) \mod p$$ $$mul: (a\times b) \mod p$$ ...
2
votes
1answer
55 views

Has a finite group, generated by $a,b$ always a relation of the form $1=a b a^{\alpha} b^{\beta}$?

Question as stated in the title: Has a finite group, generated by $a,b$ always a relation of the form $1=a b a^{\alpha} b^{\beta}$? If not, can you give me a counterexample? Thanks
1
vote
1answer
41 views

Show that $|Aut(G)|<n^{\log_2(n)}$ where $G$ is finite

Let $G$ be a finite group of order $n>1$. Show that $|Aut($G$)|<n^{\log_2(n)}$. The 'only' thing I know is the group of inner automorphisms is isomorphic to $G/Z(G)$ and by definition, ...
1
vote
0answers
37 views

Exact sequences of $1 \to A \to SO(N) \to B \to 1$, special orthogonal group

Inspired by the nice post and this, apart from SU(N), now I am particularly looking into the exact sequences of SO(N), but I will like to loosen the conditions of the previous post, Q1. $$1 \to A ...
7
votes
1answer
118 views

Prove: if $a,b\in G$ commute with probability $>5/8$, then $G$ is abelian

Suppose that $G$ is a finite group. If $P( ab=ba ) >5/8$, prove $G$ is abelian.
1
vote
2answers
43 views

Let $G=\mathbb Z_4 \times \mathbb Z_2$. Find all $H$ subgroups of $G$ of order 2, so that $G / H$ is cyclic.

Let $G=\mathbb Z_4 \times \mathbb Z_2$. Find all $H$ subgroups of $G$ of order 2, so that $G / H$ is cyclic. I tried going brute force, and realized there are 3 candidates: $H_1=\{(0,0), (2,0) \}$ ...
3
votes
0answers
51 views

Show $\exists x\in\mathbb{R}^2$ such that $g(x)\neq x$ for all $1\neq g\in G$ for a movement [closed]

Let $G\subset B_2$ ($B_n$ is a movement $\beta:\mathbb{R}^n\to\mathbb{R}^n$ which preserves the norm $||v-w||=||\beta v-\beta w||$ and is also bijective) a discrete subgroup (The subgroup ...
2
votes
1answer
26 views

Conjugacy classes of two element in a group with cyclic Sylow subgroup

Let $G$ be a finite group such that Sylow $p$-subgroup $G$ has order $p$. Let $x$ and $y$ be two elements of order $p$. Is true that $x$ and $y$ are conjugate in $G$?
1
vote
0answers
61 views

On subgroups of a finite abelian group

Let $G$ be a finite abelian group such that $G$ is of odd order or decomposition of the sylow 2-subgroups of $G$ there is at least two cyclic direct factors of maximal order. Then prove or disprove ...
0
votes
2answers
186 views

Total number of non isomorphic groups of order 122.

Let $G$ be group of order $122 = 61 \cdot 2 = p \cdot q$ , where $p < q$ are primes. I know that there exists a unique non abelian group of order $pq$ and one abelian non isomorphic group of order ...
4
votes
3answers
95 views

Show that number of solutions satisfying $x^5=e$ is a multiple of 4?

In a finite group, show that the number of nonidentity elements that satisfy the equation $x^5 = e$ is a multiple of 4. I think that besides $ x, x^2, x^3, x^4 $ also satisfies the given equation but ...
3
votes
1answer
48 views

Characterizing finite non-abelian groups in which every subgroup is abelian

How to prove: A non-abelian finite group in which every subgroup is abelian has order divisible by at most two primes.
2
votes
3answers
43 views

$SL(V)$ and $PSL(V)$ act $k$-transitively on the space of all $1$-dimensional subspaces.

A group $G$ acts $k$-transitive on some set $X$ if for every two $k$-tupels $(x_1, \ldots, x_k)$ and $(y_1, \ldots, y_k)$ there exists some $g \in G$ such that $$ g\cdot x_1 = y_1, \ldots, g\cdot ...
5
votes
1answer
59 views

Determining a group $G$ by looking at the number of homomorphisms $H\to G$

I read somewhere that, given a finite group $G$, its structure is completely determined from the knowledge of the values of $|\{H\to G\}|$ (the number of homomorphisms from $H$ to $G$) as $H$ varies ...
0
votes
1answer
32 views

Isomorphism of Groups (C.S.I.R)

Suppose $(F , +, .)$ is the finite Field with 9 elements. Let $G = (F , +)$ and H = (F \ {0}, .) denotes the underlying additive and multiplicative groups respectively, Then $ G \cong \mathbb Z_3 ...
3
votes
2answers
38 views

Index and normal subgroups

I want to show the following. For an infinite group G with only two normal subgroups (G and {e}) holds: There does not exist a non-trivial subgroup of G with finite index. I think i should prove ...
0
votes
2answers
43 views

Extension field on $\mathbb Q$

Pick the correct statements $\mathbb Q (\sqrt 2)$ and $\mathbb Q(i)$ are isomorphic as $\mathbb Q$ vector space. $\mathbb Q (\sqrt 2)$ and $\mathbb Q(i)$ are isomorphic as fielods. $Gal_{\mathbb ...
1
vote
1answer
41 views

2-Frobenius Groups of order 25920

A group $G$ is called a 2-Frobenius group if $G=ABC$, where $A$ and $AB$ are normal subgroups of $G$, $AB$ is a Frobenius group with kernel $A$ and complement $B$ and $BC$ is a Frobenius group with ...
0
votes
1answer
20 views

Conjugating a permutation

I am trying to see that two permutations are conjugate exactly when they have the same cycle decomposition. I fail to see that $$r(i_1,i_2,\dots,i_k)r^{−1}=(r(i_1),r(i_2),\dots,r(i_k))$$ Any thoughts ...
0
votes
1answer
26 views

Are augmented algebra maps of group algebras group homomorphisms?

Given a finite group $G$ and a field $k$, we can form the group algebra $kG$ with basis the elements of $G$. There is a natural augmentation $\varepsilon\colon G\to k$ that sends an element to the sum ...
1
vote
1answer
14 views

Group of permutations and disjoint cycles.

Two cycle $[i_1,..., i_r]$ and $[j_1,...,j_s]$ are said disjoint if no integer $i_\eta$ is equal to any integer $j_{\mu}$. Prove that a pemutation is equal to a product of disjoint cycles. I was ...
5
votes
3answers
50 views

Show that: $\frac{D_n}{\langle a\rangle}\simeq\mathbb{Z_2}$

Show that: $$\frac{D_n}{\langle a\rangle}\simeq\mathbb{Z_2}$$ where $D_n$ is dihedral group and $a$ is generator of order $n$.
9
votes
0answers
117 views

Is there any simple proof for this?

Let $G$ be a group with $|G:Z(G)|=n$ then $\phi(x)=x^n$ is a homomorphism from $G$ to $Z(G)$. I guess it has a proof using transfer theory, I wonder whether it has an elemantary proof or not. ...
1
vote
1answer
30 views

Understanding the elements in groups(modulo, cyclic and other(?))

Question exactly as given on past exam: Consider the group $G=(\mathbb{Z}/15\mathbb{Z})^\times$ (under multiplication). Let $H$ be the subgroup generated by $2$ (that is, $H = \langle 2 \rangle$). ...
1
vote
2answers
25 views

Having trouble understanding phrasing.

I am having a little trouble understanding the following: "If $p_1, \ldots, p_k$ be the list of distinct primes dividing the product $mn,$ then we can factor $m$ and $n$ as $m=p_1^{r_1} \cdots ...
0
votes
0answers
60 views

Simple groups of order 7920

I would like to prove that every finite simple group of order 7920 is isomorphic to $M_{11}$. I would be appreciated if someone can give me some points.
3
votes
4answers
109 views

Mapping from $\{1,\ldots,n!\}$ to the symmetric group $S_n$

Is there an easy known bijective mapping formula between the set $\{1,\ldots,n!\}$ and the symmetric group $S_n$? I want to pick a number $k \in \{1,\ldots ,n!\}$ and assign a unique permutation of ...
5
votes
1answer
61 views

Recovering a group action from sizes of orbits of individual elements

Let $G$ be a group (say, finite) and let it act on a set $X$ (say, also finite). For every element $g \in G$, we can consider its action on $X$. My rather vague question is What information about ...
1
vote
1answer
59 views

FC groups with infinite derived subgroup which are not constructed by direct product of finite groups.

A group $G$ is called FC (finite conjugacny) if every conjugacy class $C$ of $G$ has a finite order. It is called FD if the derived subgroup (constructed by commutators) is finite. It is clear that ...
3
votes
3answers
30 views

Derived Subgroup and Factor Groups

Let $N \unlhd G$, does it holds that $(G/N)' = G'N /N$, or more generally for any subgroup $H \le G$ we have $(HN/N)' = H'N/N$? Does anyone has a proof of this fact? PS: Side-Question: Is it wrong to ...
1
vote
0answers
34 views

Sort-of-multiplicative functions on the group algebra

Let $G$ be a finite group. Which functions $f:G \to \mathbf{C}$ obey the equation $$ \sum_{g \in C_1,h \in C_2} f(gh) = \left(\sum_{g \in C_1} f(g) \right)\left(\sum_{h \in C_2} f(h)\right) $$ for ...
1
vote
0answers
24 views

Poincare polynomial of a finite $G$-module with $G$ being a $p$-group

Recently, I've been reading Shatz's book, profinite groups, arithmetic, and geometry. Let $G$ be a finite $p$-group and $A$ a finite $G$-module such that $pA=(0)$. In the proof of Theorem 19 (p.82), ...
0
votes
0answers
21 views

Using a theorem to find the center of a $p$-sylow subgroup of simple group

I think that we can use the Theorem 5.3.3 of Carter's Simple book to find the $Z(P)$, where $P$ is a Sylow $p$-subgroup of a Chevalley group over a finite field of characteristic $p$ as will be shown ...
1
vote
1answer
33 views

Is $H/H_0 = HG/H_0G$?

If I have a quotient group $H/H_0$ and another group $G$ such that $H_0G\unlhd HG$. Is it then true that $H/H_0 = HG/H_0G$?